Optimal. Leaf size=464 \[ -\frac {(a-i b)^3 (i A+B-i C) \sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {(a+i b)^3 (i A-B-i C) \sqrt {c+i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {2 \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (40 a^3 C d^3-6 a^2 b d^2 (16 c C-45 B d)+9 a b^2 d \left (8 c^2 C-14 B c d+35 (A-C) d^2\right )-b^3 \left (16 c^3 C-24 B c^2 d+42 c (A-C) d^2+105 B d^3\right )\right ) (c+d \tan (e+f x))^{3/2}}{315 d^4 f}+\frac {2 b \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{3/2}}{105 d^3 f}-\frac {2 (2 b c C-3 b B d-2 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{21 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f} \]
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Rubi [A]
time = 1.41, antiderivative size = 464, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 8, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.170, Rules used = {3728, 3718,
3711, 3609, 3620, 3618, 65, 214} \begin {gather*} \frac {2 (c+d \tan (e+f x))^{3/2} \left (40 a^3 C d^3-6 a^2 b d^2 (16 c C-45 B d)+9 a b^2 d \left (35 d^2 (A-C)-14 B c d+8 c^2 C\right )-\left (b^3 \left (42 c d^2 (A-C)-24 B c^2 d+105 B d^3+16 c^3 C\right )\right )\right )}{315 d^4 f}+\frac {2 \left (a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 b \tan (e+f x) (c+d \tan (e+f x))^{3/2} \left (21 b d^2 (a B+A b-b C)+4 (b c-a d) (-2 a C d-3 b B d+2 b c C)\right )}{105 d^3 f}-\frac {(a-i b)^3 \sqrt {c-i d} (i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {(a+i b)^3 \sqrt {c+i d} (i A-B-i C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}-\frac {2 (-2 a C d-3 b B d+2 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{21 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 3609
Rule 3618
Rule 3620
Rule 3711
Rule 3718
Rule 3728
Rubi steps
\begin {align*} \int (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx &=\frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}+\frac {2 \int (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)} \left (-\frac {3}{2} (2 b c C-a (3 A-C) d)+\frac {9}{2} (A b+a B-b C) d \tan (e+f x)-\frac {3}{2} (2 b c C-3 b B d-2 a C d) \tan ^2(e+f x)\right ) \, dx}{9 d}\\ &=-\frac {2 (2 b c C-3 b B d-2 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{21 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}+\frac {4 \int (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)} \left (\frac {3}{4} \left (a^2 (21 A-13 C) d^2+4 b^2 c (2 c C-3 B d)-a b d (16 c C+9 B d)\right )+\frac {63}{4} \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2 \tan (e+f x)+\frac {3}{4} \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right ) \tan ^2(e+f x)\right ) \, dx}{63 d^2}\\ &=\frac {2 b \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{3/2}}{105 d^3 f}-\frac {2 (2 b c C-3 b B d-2 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{21 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}-\frac {8 \int \sqrt {c+d \tan (e+f x)} \left (-\frac {3}{8} \left (5 a^3 (21 A-13 C) d^3+18 a b^2 c d (4 c C-7 B d)-3 a^2 b d^2 (32 c C+15 B d)-2 b^3 c \left (8 c^2 C-12 B c d+21 (A-C) d^2\right )\right )-\frac {315}{8} \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) d^3 \tan (e+f x)-\frac {3}{8} \left (40 a^3 C d^3-6 a^2 b d^2 (16 c C-45 B d)+9 a b^2 d \left (8 c^2 C-14 B c d+35 (A-C) d^2\right )-b^3 \left (16 c^3 C-24 B c^2 d+42 c (A-C) d^2+105 B d^3\right )\right ) \tan ^2(e+f x)\right ) \, dx}{315 d^3}\\ &=\frac {2 \left (40 a^3 C d^3-6 a^2 b d^2 (16 c C-45 B d)+9 a b^2 d \left (8 c^2 C-14 B c d+35 (A-C) d^2\right )-b^3 \left (16 c^3 C-24 B c^2 d+42 c (A-C) d^2+105 B d^3\right )\right ) (c+d \tan (e+f x))^{3/2}}{315 d^4 f}+\frac {2 b \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{3/2}}{105 d^3 f}-\frac {2 (2 b c C-3 b B d-2 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{21 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}-\frac {8 \int \sqrt {c+d \tan (e+f x)} \left (\frac {315}{8} \left (3 a^2 b B-b^3 B-a^3 (A-C)+3 a b^2 (A-C)\right ) d^3-\frac {315}{8} \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) d^3 \tan (e+f x)\right ) \, dx}{315 d^3}\\ &=\frac {2 \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (40 a^3 C d^3-6 a^2 b d^2 (16 c C-45 B d)+9 a b^2 d \left (8 c^2 C-14 B c d+35 (A-C) d^2\right )-b^3 \left (16 c^3 C-24 B c^2 d+42 c (A-C) d^2+105 B d^3\right )\right ) (c+d \tan (e+f x))^{3/2}}{315 d^4 f}+\frac {2 b \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{3/2}}{105 d^3 f}-\frac {2 (2 b c C-3 b B d-2 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{21 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}-\frac {8 \int \frac {-\frac {315}{8} d^3 \left (a^3 (A c-c C-B d)-3 a b^2 (A c-c C-B d)-3 a^2 b (B c+(A-C) d)+b^3 (B c+(A-C) d)\right )-\frac {315}{8} d^3 \left (3 a^2 b (A c-c C-B d)-b^3 (A c-c C-B d)+a^3 (B c+(A-C) d)-3 a b^2 (B c+(A-C) d)\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{315 d^3}\\ &=\frac {2 \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (40 a^3 C d^3-6 a^2 b d^2 (16 c C-45 B d)+9 a b^2 d \left (8 c^2 C-14 B c d+35 (A-C) d^2\right )-b^3 \left (16 c^3 C-24 B c^2 d+42 c (A-C) d^2+105 B d^3\right )\right ) (c+d \tan (e+f x))^{3/2}}{315 d^4 f}+\frac {2 b \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{3/2}}{105 d^3 f}-\frac {2 (2 b c C-3 b B d-2 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{21 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}+\frac {1}{2} \left ((a-i b)^3 (A-i B-C) (c-i d)\right ) \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx+\frac {1}{2} \left ((a+i b)^3 (A+i B-C) (c+i d)\right ) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx\\ &=\frac {2 \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (40 a^3 C d^3-6 a^2 b d^2 (16 c C-45 B d)+9 a b^2 d \left (8 c^2 C-14 B c d+35 (A-C) d^2\right )-b^3 \left (16 c^3 C-24 B c^2 d+42 c (A-C) d^2+105 B d^3\right )\right ) (c+d \tan (e+f x))^{3/2}}{315 d^4 f}+\frac {2 b \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{3/2}}{105 d^3 f}-\frac {2 (2 b c C-3 b B d-2 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{21 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}-\frac {\left (i (a+i b)^3 (A+i B-C) (c+i d)\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{2 f}+\frac {\left ((a-i b)^3 (A-i B-C) (i c+d)\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{2 f}\\ &=\frac {2 \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (40 a^3 C d^3-6 a^2 b d^2 (16 c C-45 B d)+9 a b^2 d \left (8 c^2 C-14 B c d+35 (A-C) d^2\right )-b^3 \left (16 c^3 C-24 B c^2 d+42 c (A-C) d^2+105 B d^3\right )\right ) (c+d \tan (e+f x))^{3/2}}{315 d^4 f}+\frac {2 b \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{3/2}}{105 d^3 f}-\frac {2 (2 b c C-3 b B d-2 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{21 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}-\frac {\left ((a+i b)^3 (A+i B-C) (c+i d)\right ) \text {Subst}\left (\int \frac {1}{-1+\frac {i c}{d}-\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{d f}-\frac {\left ((i a+b)^3 (A-i B-C) (i c+d)\right ) \text {Subst}\left (\int \frac {1}{-1-\frac {i c}{d}+\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{d f}\\ &=-\frac {(a-i b)^3 (i A+B-i C) \sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}-\frac {(i a-b)^3 (A+i B-C) \sqrt {c+i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {2 \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (40 a^3 C d^3-6 a^2 b d^2 (16 c C-45 B d)+9 a b^2 d \left (8 c^2 C-14 B c d+35 (A-C) d^2\right )-b^3 \left (16 c^3 C-24 B c^2 d+42 c (A-C) d^2+105 B d^3\right )\right ) (c+d \tan (e+f x))^{3/2}}{315 d^4 f}+\frac {2 b \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{3/2}}{105 d^3 f}-\frac {2 (2 b c C-3 b B d-2 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{21 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(1232\) vs. \(2(464)=928\).
time = 6.29, size = 1232, normalized size = 2.66 \begin {gather*} \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}+\frac {2 \left (-\frac {3 (2 b c C-3 b B d-2 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{7 d f}+\frac {2 \left (\frac {3 b \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{3/2}}{10 d f}-\frac {2 \left (\frac {2 \left (-\frac {15}{8} a d \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right )+b \left (-\frac {315}{8} \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+\frac {3}{4} c \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right )\right )\right ) (c+d \tan (e+f x))^{3/2}}{3 d f}+\frac {i \left (-\frac {15}{8} a d \left (a^2 (21 A-13 C) d^2+4 b^2 c (2 c C-3 B d)-a b d (16 c C+9 B d)\right )+\frac {3}{4} b c \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right )+\frac {15}{8} a d \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right )+\frac {5}{2} i d \left (\frac {63}{4} a \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2+\frac {3}{4} b \left (a^2 (21 A-13 C) d^2+4 b^2 c (2 c C-3 B d)-a b d (16 c C+9 B d)\right )-\frac {3}{4} b \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right )\right )-b \left (-\frac {315}{8} \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+\frac {3}{4} c \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right )\right )\right ) \left (\frac {2 (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{-c+i d}+2 \sqrt {c+d \tan (e+f x)}\right )}{2 f}-\frac {i \left (-\frac {15}{8} a d \left (a^2 (21 A-13 C) d^2+4 b^2 c (2 c C-3 B d)-a b d (16 c C+9 B d)\right )+\frac {3}{4} b c \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right )+\frac {15}{8} a d \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right )-\frac {5}{2} i d \left (\frac {63}{4} a \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2+\frac {3}{4} b \left (a^2 (21 A-13 C) d^2+4 b^2 c (2 c C-3 B d)-a b d (16 c C+9 B d)\right )-\frac {3}{4} b \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right )\right )-b \left (-\frac {315}{8} \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+\frac {3}{4} c \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right )\right )\right ) \left (\frac {2 (c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{-c-i d}+2 \sqrt {c+d \tan (e+f x)}\right )}{2 f}\right )}{5 d}\right )}{7 d}\right )}{9 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3065\) vs.
\(2(424)=848\).
time = 0.68, size = 3066, normalized size = 6.61
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(3066\) |
default | \(\text {Expression too large to display}\) | \(3066\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \tan {\left (e + f x \right )}\right )^{3} \sqrt {c + d \tan {\left (e + f x \right )}} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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